The LMAPS Using Polynomial Basis Functions for Near-Singular Problems (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we combine the Local Method of Approximate Particular Solutions(LMAPS) with polynomial basis functions to solve near-singular problems. Dueto the unique feature of the local approach, the LMAPS is capable of capturing therapid variation of the solution. Polynomial basis functions can become very unstablewhen the order of the polynomial becomes large. However, since the LMAPS is a localmethod, the order does not need to be very high; an order of 5 can achieve sufficientaccuracy. In order to show the effectiveness of the LMAPS using polynomial basisfunctions for solving near singular problems, we compare the results with the LMAPSusing radial basis functions (RBFs). The advantage of using polynomials as a basisrather than RBFs is that finding an appropriate shape parameter is not necessary.